Rattleback dynamics and its reversal time of rotation

A rattleback is a rigid, semielliptic toy which exhibits unintuitive behavior; when it is spun in one direction, it soon begins pitching and stops spinning, then it starts to spin in the opposite direction, but in the other direction, it seems to spin just steadily. This puzzling behavior results from the slight misalignment between the principal axes for the inertia and those for the curvature; the misalignment couples the spinning with the pitching and the rolling oscillations. It has been shown that under the no-slip condition and without dissipation the spin can reverse in both directions, and Garcia and Hubbard obtained the formula for the time required for the spin reversal $t_r$ [Proc. R. Soc. Lond. A 418, 165 (1988)]. In this work, we reformulate the rattleback dynamics in a physically transparent way and reduce it to a three-variable dynamics for spinning, pitching, and rolling. We obtain an expression of the Garcia-Hubbard formula for $t_r$ by a simple product of four factors: (1) the misalignment angle, (2) the difference in the inverses of inertia moment for the two oscillations, (3) that in the radii for the two principal curvatures, and (4) the squared frequency of the oscillation. We perform extensive numerical simulations to examine validity and limitation of the formula, and find that (1) the Garcia-Hubbard formula is good for both spinning directions in the small spin and small oscillation regime, but (2) in the fast spin regime especially for the steady direction, the rattleback may not reverse and shows a rich variety of dynamics including steady spinning, spin wobbling, and chaotic behavior reminiscent of chaos in a dissipative system.

Figure 1

(a) A commercially available rattleback made of plastic. (b) Notations of the rattleback. (c) A schematic illustration of the shell-dumbbell model.

Figure 2

A body-fixed coordinate viewed from below. The dashed lines indicate the principal directions of curvature, rotated by $\xi$ from the principal axes of inertia (the $x\text{−}y$ axes).

Figure 3

(a) Cumulative distributions of the pitch and the roll frequencies for the parameter set SD in Table 1; ${\omega }_p\text{and}{\omega }_r$ of Eq. (23) and their zeroth order approximation ${\omega }_{p0}$ and ${\omega }_{r0}$ by Eqs. (28) and (29) are shown. The inset shows the cumulative distribution of ${\omega }_p/{\omega }_r$. The number of samples is ${10}^6$.

Figure 4

(a) Cumulative distributions of the asymmetric torque coefficients $K_p$ and $K_r$ for SD (Table 1). The number of samples is ${10}^5$. (b) A 2D color plot for the distribution of $(K_p,K_r)$. The color code shown is in the logarithmic scale for the relative frequency $P(K_p,K_r)$, i.e., $-9\le {log}_{10}P(K_p,K_r)\le 0$. The number of samples is ${10}^8$.

Figure 5

A typical spin evolution and the corresponding ${\omega }_x$ and ${\omega }_y$ for GH (Table 1). (a) The case of the initial spin in the unsteady direction. The initial condition is specified by Eqs. (68, 69, 70) with $n_i=0.1\tilde{\omega }$ and $|{\omega }_{xy0}|=0.01\tilde{\omega }$. (b) The case of the initial spin in the steady direction with $n_i=-0.1\tilde{\omega }$ and $|{\omega }_{xy0}|=0.01\tilde{\omega }$. The dashed lines in (a-1) and (b-1) show Garcia and Hubbard's solution $\overline{n}\left(t\right)$ given by Eqs. (55) and (57), respectively.

Figure 6

(a) Time for reversal of the unsteady direction $t_{ru}$ for the parameter set SD (Table 1) as a function of the asymmetric torque coefficient $K_p$ in the logarithmic scale. The error bars indicate one standard deviation of 1000 samples for each data point. The solid lines are $t_{rGH+}$ given by Eq. (56), calculated using the mean values of $n_0$. (b) A typical spin evolution with $n_i=0.5\tilde{\omega },\left|{\omega }_{xy0}\right|=0.01\tilde{\omega }$. The parameter set GH is used.

Figure 7

(a) Fractions of types R, SS, and SW for the steady direction for eight values of $K_r$ with various initial conditions $|{\omega }_{xy0}|\text{and}{n}_i$. Parameters are randomly chosen from SD (Table 1). The number of the samples is 1000 for each $K_r$. Filled triangles show the fractions of samples whose $|n_{c1}|$ is smaller than $|n_i|$. (b) Typical spin evolutions of a Type SS sample (b-1) and a Type SW sample (b-2), along with an example of “chaotic” oscillation (b-3) found for $K_r=0.0041$ with $n_i=-0.5\tilde{\omega }$.

Figure 8

Time for reversal $t_{rs}$ for the steady direction as a function of $K_r$ in the logarithmic scale. Each data point represents the average with the standard deviation of Type R samples out of 1000 simulations from the parameter set SD (Table 1).

Figure 9

Three types of spin behaviors after the first reversal period in the small spin regime with $n_i=0.1\tilde{\omega },|{\omega }_{xy0}|=0.01\tilde{\omega }$. (a) A quasiperiodic behavior with the parameter set GH $\left(\theta =0.6429\right)$, (b) a chaotic behavior with $\theta =0.82$, (c) a quasiperiodic behavior with a period shorter than the first one with $\theta =0.9$. All the other parameters for (b) and (c) are the same as GH. The dashed lines show the spin evolutions for the corresponding simplified dynamics, where ${\overline{E}}_p\left(0\right)={Ma}^2\left[\alpha \right(|{\omega }_{xy0}|cos{\psi }_p{)}^2+\beta \left(\right|{\omega }_{xy0}|sin{\psi }_p{)}^2]/2,{\overline{E}}_r\left(0\right)=3\times {10}^{-5}{Ma}^2{\tilde{\omega }}^2$, and $\overline{n}\left(0\right)=n_i$.